The DM is being too harsh.
While the crafting rules in Xanathar's Guide to Everything put it on the DM's shoulders to decide what tools are required, nothing in those instructions ever implies that multiple tool sets should be required.
The optional rules for crafting an item say this (XGE, p. 128):
A character needs to be proficient with the tools needed to craft an item and have access to the appropriate equipment. Everyone who collaborates needs to have the appropriate tool proficiency.
This passage uses the singular "tool proficiency", not "proficiencies", and the sample table implies that smith's tools are sufficient to make almost any armor. It seems overly limiting to me to require two separate crafting skills before you're able to make basic scale armor. How many characters even have two tool proficiencies, let alone these two in particular?
Since Serpent Scale Armor (Candlekeep Mysteries, p. 98) is magical, there's even yet another option:
To complete a magic item, a character also needs whatever tool proficiency is appropriate, as for crafting a nonmagical object, or proficiency in the Arcana skill.
I assume you can't go the Arcana route, but again -- note that it uses the singular 'tool proficiency'. It doesn't even consider the idea of needing to have two separate tool sets just because the armor has leather parts.
Your DM can do whatever they want to, but this seems like an absurd requirement to put on the players, unless the DM is just using this as a pretext to prevent you from making items.